IE 423 PROJECT PART 1

PART 1

1. We have used “jpeg” package as stated in the document of project explanation. readJPEG() function in the jpeg package stores an image as an object in R. Our object name is “img”. You should change your image directory if you want to run our code yourself.

library(jpeg)
img <- readJPEG("C:/Users/ilker zeybek/Desktop/423part1.jpg", native = FALSE)

2. Our image is stored as a dot matrix. They are representing pixel values of our image in the stored 512x512x3 matrix object called “img”. We have displayed our image using rasterImage() function.

plot(NA,xlim=c(0,nrow(img)),ylim=c(0,ncol(img)))
rasterImage(img,0,0,nrow(img),ncol(img))

We have created objects called “a”,“b”, and “c” for storing the first, second, and the third channels of our image, respectively. Then plotted channels using rasterImage() function and showed them in a single plot.

a <- img[,,1]
b <- img[,,2]
c <- img[,,3]
par(mfrow=c(1,3))
plot(NA,xlim=c(0,nrow(a)),ylim=c(0,ncol(a)))
rasterImage(a,0,0,nrow(a),ncol(a))
plot(NA,xlim=c(0,nrow(b)),ylim=c(0,ncol(b)))
rasterImage(b,0,0,nrow(b),ncol(b))
plot(NA,xlim=c(0,nrow(c)),ylim=c(0,ncol(c)))
rasterImage(c,0,0,nrow(c),ncol(c))

3. We have stored means of columns of channels in the objects named “x”,“y”, and “z”, respectively. After storing them, we used par() function to plot() the array of column means in a single plot.

x <- colMeans(a)
y <- colMeans(b)
z <- colMeans(c)
par(mfrow=c(1,1))
plot(x, type = "l", col="red",xlab = "Columns" , ylab = "Mean", ylim = c(0.3,1))
lines(y,col="Green")
lines(z,col="Blue")

4. We have created objects named “k”, “l”, “m” to operate on our channel matrices. We have decided to subtract bottom half of the channels from their top half. We have manipulated our “a”, “b”, and “c” matrix rows and stored the new half images in k and l objects. m object is used for the final matrix that came from subtraction of k and l matrices. Since we can not control which cell value is bigger than the other, we took absolute value of the final matrix m in order to display it with rasterImage() function.(k and l objects are for channel 1, k2 and l2 objects are for channel 2, k3 and l3 objects for channel 3)

k <- a[1:256,1:512]
l <- a[257:512,1:512]
m <- k-l
plot(NA,xlim=c(0,nrow(m)),ylim=c(0,ncol(m)))
rasterImage(abs(m),0,0,nrow(m),ncol(m))

k2 <- b[1:256,1:512]
l2 <- b[257:512,1:512]
m2 <- k2-l2
plot(NA,xlim=c(0,nrow(m2)),ylim=c(0,ncol(m2)))
rasterImage(abs(m2),0,0,nrow(m2),ncol(m2))

k3 <- c[1:256,1:512]
l3 <- c[257:512,1:512]
m3 <- k3-l3
plot(NA,xlim=c(0,nrow(m3)),ylim=c(0,ncol(m3)))
rasterImage(abs(m3),0,0,nrow(m3),ncol(m3))

5. In order to apply median filtering, we have used “imagine” package for filtering channels of our image. Imaging package have medianFilter() function which needs arguments as object, specified window, and how many times you would apply the filter. For example, medianFilter(img,radius=10,times=1). This example executes 10x10 window of median filter to img object 1 time. We have stored our median filtered images in “fivemedianfiltered”, “elevenmedianfiltered”, and “thirtyonemedianfiltered” objects (at the end of the object names there are indicators named a, b, and c which represents channels 1, 2, and 3, respectively). We have displayed median filtered images with using rasterImage() function.

library(imagine)
fivemedianfiltereda <- medianFilter(a,radius=5,times=1)
plot(NA,xlim=c(0,nrow(fivemedianfiltereda)),ylim=c(0,ncol(fivemedianfiltereda)))
rasterImage(fivemedianfiltereda,0,0,nrow(fivemedianfiltereda),ncol(fivemedianfiltereda))

fivemedianfilteredb <- medianFilter(b,radius=5,times=1)
plot(NA,xlim=c(0,nrow(fivemedianfilteredb)),ylim=c(0,ncol(fivemedianfilteredb)))
rasterImage(fivemedianfilteredb,0,0,nrow(fivemedianfilteredb),ncol(fivemedianfilteredb))

fivemedianfilteredc <- medianFilter(c,radius=5,times=1)
plot(NA,xlim=c(0,nrow(fivemedianfilteredc)),ylim=c(0,ncol(fivemedianfilteredc)))
rasterImage(fivemedianfilteredc,0,0,nrow(fivemedianfilteredc),ncol(fivemedianfilteredc))

elevenmedianfiltereda <- medianFilter(a,radius=11,times=1)
plot(NA,xlim=c(0,nrow(elevenmedianfiltereda)),ylim=c(0,ncol(elevenmedianfiltereda)))
rasterImage(elevenmedianfiltereda,0,0,nrow(elevenmedianfiltereda),ncol(elevenmedianfiltereda))

elevenmedianfilteredb <- medianFilter(b,radius=11,times=1)
plot(NA,xlim=c(0,nrow(elevenmedianfilteredb)),ylim=c(0,ncol(elevenmedianfilteredb)))
rasterImage(elevenmedianfilteredb,0,0,nrow(elevenmedianfilteredb),ncol(elevenmedianfilteredb))

elevenmedianfilteredc <- medianFilter(c,radius=11,times=1)
plot(NA,xlim=c(0,nrow(elevenmedianfilteredc)),ylim=c(0,ncol(elevenmedianfilteredc)))
rasterImage(elevenmedianfilteredc,0,0,nrow(elevenmedianfilteredc),ncol(elevenmedianfilteredc))

thirtyonemedianfiltereda <- medianFilter(a,radius=31,times=1)
plot(NA,xlim=c(0,nrow(thirtyonemedianfiltereda)),ylim=c(0,ncol(thirtyonemedianfiltereda)))
rasterImage(thirtyonemedianfiltereda,0,0,nrow(thirtyonemedianfiltereda),ncol(thirtyonemedianfiltereda))

thirtyonemedianfilteredb <- medianFilter(b,radius=31,times=1)
plot(NA,xlim=c(0,nrow(thirtyonemedianfilteredb)),ylim=c(0,ncol(thirtyonemedianfilteredb)))
rasterImage(thirtyonemedianfilteredb,0,0,nrow(thirtyonemedianfilteredb),ncol(thirtyonemedianfilteredb))

thirtyonemedianfilteredc <- medianFilter(c,radius=31,times=1)
plot(NA,xlim=c(0,nrow(thirtyonemedianfilteredc)),ylim=c(0,ncol(thirtyonemedianfilteredc)))
rasterImage(thirtyonemedianfilteredc,0,0,nrow(thirtyonemedianfilteredc),ncol(thirtyonemedianfilteredc))

Median filtering is a pre-processing tool used for removing salt and pepper noise in the image. In our project, bigger window sizes of median filter caused significant loss in signal/noise ratio. Certain color shifts looking like cracks in the table started to blend in more with the rest of the colors in the table. As we have expected before starting to median filtering, bigger window size is unreliable in median filtering. The more data you have in the window, the more you lose your reliable pixels in the image.

PART 2

1. We have used “jpeg” package to store our image and drew histogram of the pixels in the image using hist() function.

part2img <- readJPEG("C:/Users/ilker zeybek/Desktop/423part1grayscale.jpg", native = FALSE)
hist(part2img)

As can be seen in the histogram, it is a bell shaped curve. We assumed it is normally distributed.

2. We took a sample from our “part2img” matrix and tested whether we are right or wrong in our assumption of probability distribution using “Shapiro-Wilk Test”.

sample <- sample(part2img,1000,replace = FALSE)
shapiro.test(sample)

## 
##  Shapiro-Wilk normality test
## 
## data:  sample
## W = 0.92639, p-value < 2.2e-16

P-value is sufficiently small so we conclude that our assumption about probability distribution, which was assumed normally distributed, is true.

To estimate parameters of normally distributed population, we used mean of the sample and standard deviation of the sample.(n=1000)

mean(sample)

## [1] 0.584949

sd(sample)

## [1] 0.06107764

3. To find and upperlimit and lowerlimit of 0.001 probability, we should use inverse of cumulative distribution function, which qnorm() function in R.

upperlimit <- qnorm(0.999,mean(sample),sd(sample))
lowerlimit <- qnorm(0.001,mean(sample),sd(sample))

Upper limit of pixel values

upperlimit

## [1] 0.7736931

Lower limit of pixel values

lowerlimit

## [1] 0.3962049

Pixel values identified over the upperlimit

part2img[part2img > upperlimit]

## numeric(0)

Pixel values identified below the lowerlimit

part2img[part2img < lowerlimit]

##    [1] 0.3960784 0.3921569 0.3686275 0.3686275 0.3725490 0.3843137
##    [7] 0.3686275 0.3450980 0.3647059 0.3686275 0.3568627 0.3921569
##   [13] 0.3803922 0.3882353 0.3960784 0.3803922 0.3803922 0.3607843
##   [19] 0.3647059 0.3607843 0.3450980 0.3411765 0.3764706 0.3882353
##   [25] 0.3607843 0.3254902 0.2941176 0.3372549 0.3568627 0.3607843
##   [31] 0.3764706 0.3921569 0.3803922 0.3803922 0.3843137 0.3882353
##   [37] 0.3960784 0.3921569 0.3843137 0.3843137 0.3882353 0.3803922
##   [43] 0.3529412 0.3607843 0.3686275 0.3960784 0.3764706 0.3607843
##   [49] 0.3960784 0.3843137 0.3764706 0.3490196 0.3215686 0.3137255
##   [55] 0.3176471 0.2901961 0.3098039 0.3294118 0.3215686 0.3176471
##   [61] 0.3411765 0.3764706 0.3921569 0.3960784 0.3647059 0.3333333
##   [67] 0.3803922 0.3764706 0.3803922 0.3843137 0.3960784 0.3843137
##   [73] 0.3725490 0.3764706 0.3803922 0.3725490 0.3607843 0.3607843
##   [79] 0.3843137 0.3882353 0.3725490 0.3568627 0.3647059 0.3764706
##   [85] 0.3725490 0.3882353 0.3686275 0.3490196 0.3450980 0.3568627
##   [91] 0.3686275 0.3921569 0.3647059 0.3568627 0.3725490 0.3803922
##   [97] 0.3450980 0.3647059 0.3686275 0.3529412 0.3921569 0.3921569
##  [103] 0.3803922 0.3568627 0.3490196 0.3450980 0.3568627 0.3803922
##  [109] 0.3490196 0.3450980 0.3960784 0.3764706 0.3490196 0.3568627
##  [115] 0.3450980 0.3647059 0.3764706 0.3843137 0.3921569 0.3921569
##  [121] 0.3843137 0.3882353 0.3843137 0.3686275 0.3921569 0.3882353
##  [127] 0.3490196 0.3686275 0.3882353 0.3764706 0.3843137 0.3764706
##  [133] 0.3607843 0.3568627 0.3568627 0.3568627 0.3686275 0.3686275
##  [139] 0.3333333 0.3215686 0.3411765 0.3960784 0.3803922 0.3921569
##  [145] 0.3960784 0.3843137 0.3764706 0.3725490 0.3725490 0.3725490
##  [151] 0.3568627 0.3647059 0.3725490 0.3803922 0.3568627 0.3450980
##  [157] 0.3568627 0.3607843 0.3764706 0.3843137 0.3686275 0.3764706
##  [163] 0.3529412 0.3490196 0.3803922 0.3803922 0.3725490 0.3882353
##  [169] 0.3843137 0.3803922 0.3921569 0.3607843 0.3607843 0.3960784
##  [175] 0.3490196 0.3764706 0.3921569 0.3843137 0.3882353 0.3921569
##  [181] 0.3725490 0.3725490 0.3960784 0.3843137 0.3568627 0.3450980
##  [187] 0.3568627 0.3843137 0.3960784 0.3882353 0.3882353 0.3725490
##  [193] 0.3882353 0.3960784 0.3450980 0.3568627 0.3725490 0.3725490
##  [199] 0.3490196 0.3647059 0.3764706 0.3960784 0.3725490 0.3607843
##  [205] 0.3647059 0.3843137 0.3647059 0.3254902 0.3921569 0.3725490
##  [211] 0.3333333 0.3176471 0.3098039 0.2941176 0.3176471 0.3294118
##  [217] 0.3137255 0.3098039 0.3254902 0.3254902 0.3254902 0.3843137
##  [223] 0.3803922 0.3960784 0.3882353 0.3882353 0.3882353 0.3450980
##  [229] 0.3058824 0.3450980 0.3529412 0.3607843 0.3725490 0.3725490
##  [235] 0.3529412 0.3450980 0.3843137 0.3803922 0.3803922 0.3803922
##  [241] 0.3686275 0.3607843 0.3686275 0.3725490 0.3647059 0.3607843
##  [247] 0.3803922 0.3960784 0.3960784 0.3960784 0.3921569 0.3764706
##  [253] 0.3843137 0.3843137 0.3647059 0.3803922 0.3725490 0.3607843
##  [259] 0.3960784 0.3764706 0.3450980 0.3450980 0.3725490 0.3921569
##  [265] 0.3843137 0.3960784 0.3882353 0.3843137 0.3882353 0.3960784
##  [271] 0.3843137 0.3764706 0.3803922 0.3960784 0.3803922 0.3725490
##  [277] 0.3882353 0.3882353 0.3764706 0.3607843 0.3647059 0.3843137
##  [283] 0.3960784 0.3882353 0.3960784 0.3921569 0.3764706 0.3725490
##  [289] 0.3764706 0.3921569 0.3882353 0.3686275 0.3647059 0.3843137
##  [295] 0.3764706 0.3411765 0.3137255 0.3137255 0.3254902 0.3411765
##  [301] 0.3333333 0.3490196 0.3568627 0.3607843 0.3686275 0.3725490
##  [307] 0.3764706 0.3960784 0.3725490 0.3686275 0.3529412 0.3372549
##  [313] 0.3333333 0.3490196 0.3764706 0.3960784 0.3764706 0.3843137
##  [319] 0.3921569 0.3960784 0.3960784 0.3725490 0.3529412 0.3803922
##  [325] 0.3803922 0.3882353 0.3764706 0.3725490 0.3764706 0.3843137
##  [331] 0.3921569 0.3921569 0.3764706 0.3568627 0.3529412 0.3568627
##  [337] 0.3647059 0.3882353 0.3568627 0.3333333 0.3411765 0.3294118
##  [343] 0.3137255 0.3176471 0.3529412 0.3960784 0.3843137 0.3647059
##  [349] 0.3529412 0.3607843 0.3764706 0.3803922 0.3882353 0.3647059
##  [355] 0.3215686 0.3294118 0.3960784 0.3803922 0.3725490 0.3843137
##  [361] 0.3960784 0.3803922 0.3803922 0.3960784 0.3882353 0.3921569
##  [367] 0.3607843 0.3529412 0.3803922 0.3764706 0.3450980 0.3450980
##  [373] 0.3725490 0.3960784 0.3921569 0.3921569 0.3960784 0.3607843
##  [379] 0.3803922 0.3843137 0.3686275 0.3686275 0.3725490 0.3607843
##  [385] 0.3372549 0.3921569 0.3960784 0.3921569 0.3882353 0.3803922
##  [391] 0.3882353 0.3960784 0.3764706 0.3764706 0.3607843 0.3686275
##  [397] 0.3882353 0.3764706 0.3843137 0.3921569 0.3725490 0.3921569
##  [403] 0.3803922 0.3764706 0.3176471 0.3294118 0.3647059 0.3647059
##  [409] 0.3490196 0.3450980 0.3490196 0.3803922 0.3960784 0.3882353
##  [415] 0.3960784 0.3647059 0.3647059 0.3843137 0.3960784 0.3960784
##  [421] 0.3764706 0.3882353 0.3882353 0.3725490 0.3764706 0.3882353
##  [427] 0.3882353 0.3960784 0.3764706 0.3882353 0.3960784 0.3882353
##  [433] 0.3882353 0.3764706 0.3921569 0.3882353 0.3882353 0.3725490
##  [439] 0.3686275 0.3803922 0.3882353 0.3725490 0.3921569 0.3607843
##  [445] 0.3568627 0.3607843 0.3215686 0.3019608 0.3254902 0.3411765
##  [451] 0.3333333 0.3294118 0.3411765 0.3607843 0.3960784 0.3843137
##  [457] 0.3803922 0.3725490 0.3764706 0.3960784 0.3882353 0.3843137
##  [463] 0.3960784 0.3960784 0.3764706 0.3921569 0.3843137 0.3725490
##  [469] 0.3647059 0.3843137 0.3803922 0.3490196 0.3490196 0.3490196
##  [475] 0.3882353 0.3843137 0.3960784 0.3882353 0.3843137 0.3764706
##  [481] 0.3725490 0.3607843 0.3725490 0.3882353 0.3960784 0.3882353
##  [487] 0.3725490 0.3764706 0.3960784 0.3686275 0.3843137 0.3725490
##  [493] 0.3843137 0.3921569 0.3803922 0.3960784 0.3843137 0.3921569
##  [499] 0.3607843 0.3843137 0.3568627 0.3843137 0.3960784 0.3725490
##  [505] 0.3803922 0.3764706 0.3764706 0.3960784 0.3882353 0.3647059
##  [511] 0.3607843 0.3411765 0.3647059 0.3764706 0.3764706 0.3882353
##  [517] 0.3882353 0.3764706 0.3960784 0.3960784 0.3921569 0.3921569
##  [523] 0.3882353 0.3529412 0.3176471 0.3490196 0.3882353 0.3725490
##  [529] 0.3960784 0.3607843 0.3686275 0.3960784 0.3843137 0.3764706
##  [535] 0.3960784 0.3803922 0.3921569 0.3960784 0.3921569 0.3882353
##  [541] 0.3764706 0.3058824 0.3058824 0.3333333 0.3568627 0.3607843
##  [547] 0.3960784 0.3921569 0.3529412 0.3490196 0.3882353 0.3882353
##  [553] 0.3882353 0.3960784 0.3686275 0.3568627 0.3411765 0.3333333
##  [559] 0.3450980 0.3568627 0.3686275 0.3882353 0.3960784 0.3843137
##  [565] 0.3960784 0.3882353 0.3921569 0.3725490 0.3490196 0.3411765
##  [571] 0.3607843 0.3921569 0.3882353 0.3843137 0.3960784 0.3921569
##  [577] 0.3686275 0.3960784 0.3686275 0.3607843 0.3921569 0.3803922
##  [583] 0.3882353 0.3960784 0.3882353 0.3960784 0.3960784 0.3921569
##  [589] 0.3843137 0.3882353 0.3882353 0.3607843 0.3294118 0.3333333
##  [595] 0.3647059 0.3960784 0.3921569 0.3882353 0.3882353 0.3411765
##  [601] 0.3921569 0.3960784 0.3803922 0.3607843 0.2823529 0.2980392
##  [607] 0.3254902 0.3137255 0.3921569 0.3882353 0.3764706 0.3294118
##  [613] 0.3607843 0.3921569 0.3921569 0.3803922 0.3921569 0.3490196
##  [619] 0.3411765 0.3490196 0.3607843 0.3764706 0.3882353 0.3882353
##  [625] 0.3882353 0.3686275 0.3843137 0.3764706 0.3803922 0.3686275
##  [631] 0.3607843 0.3725490 0.3725490 0.3607843 0.3607843 0.3333333
##  [637] 0.3215686 0.3137255 0.3176471 0.3607843 0.3333333 0.3372549
##  [643] 0.3254902 0.3372549 0.3921569 0.3803922 0.3725490 0.3607843
##  [649] 0.3647059 0.3372549 0.3333333 0.3490196 0.3607843 0.3921569
##  [655] 0.3843137 0.3764706 0.3568627 0.3137255 0.3215686 0.3843137
##  [661] 0.3960784 0.3647059 0.3921569 0.3607843 0.3411765 0.3176471
##  [667] 0.2941176 0.3058824 0.3921569 0.3764706 0.3764706 0.3803922
##  [673] 0.3882353 0.3843137 0.3882353 0.3843137 0.3882353 0.3921569
##  [679] 0.3686275 0.3607843 0.3960784 0.3882353 0.3960784 0.3843137
##  [685] 0.3921569 0.3803922 0.3764706 0.3686275 0.3725490 0.3764706
##  [691] 0.3215686 0.3098039 0.3372549 0.3568627 0.3725490 0.3960784
##  [697] 0.3686275 0.3843137 0.3803922 0.3568627 0.3333333 0.3882353
##  [703] 0.3490196 0.3764706 0.3960784 0.3882353 0.3960784 0.3882353
##  [709] 0.3960784 0.3803922 0.3882353 0.3843137 0.3921569 0.3921569
##  [715] 0.3568627 0.3568627 0.3960784 0.3960784 0.3960784 0.3882353
##  [721] 0.3960784 0.3725490 0.3921569 0.3803922 0.3803922 0.3960784
##  [727] 0.3960784 0.3882353 0.3529412 0.3960784 0.3725490 0.3450980
##  [733] 0.3450980 0.3294118 0.3019608 0.3176471 0.3882353 0.3882353
##  [739] 0.3882353 0.3921569 0.3686275 0.3647059 0.3529412 0.3490196
##  [745] 0.3607843 0.3764706 0.3725490 0.3568627 0.3372549 0.3411765
##  [751] 0.3607843 0.3607843 0.3372549 0.3411765 0.3882353 0.3803922
##  [757] 0.3803922 0.3764706 0.3764706 0.3921569 0.3882353 0.3725490
##  [763] 0.3686275 0.3607843 0.3529412 0.3647059 0.3294118 0.3490196
##  [769] 0.3921569 0.3843137 0.3490196 0.3490196 0.3607843 0.3803922
##  [775] 0.3921569 0.3921569 0.3607843 0.3215686 0.3098039 0.3372549
##  [781] 0.3607843 0.3490196 0.3215686 0.3411765 0.3137255 0.3019608
##  [787] 0.3254902 0.3568627 0.3803922 0.3960784 0.3843137 0.3960784
##  [793] 0.3843137 0.3254902 0.3568627 0.3803922 0.3568627 0.3843137
##  [799] 0.3960784 0.3686275 0.3607843 0.3764706 0.3921569 0.3960784
##  [805] 0.3960784 0.3843137 0.3803922 0.3529412 0.3882353 0.3725490
##  [811] 0.3725490 0.3764706 0.3490196 0.3960784 0.3843137 0.3529412
##  [817] 0.3803922 0.3764706 0.3921569 0.3882353 0.3647059 0.3843137
##  [823] 0.3921569 0.3529412 0.3843137 0.3921569 0.3372549 0.3764706
##  [829] 0.3921569 0.3882353 0.3882353 0.3843137 0.3960784 0.3882353
##  [835] 0.3921569 0.3764706 0.3647059 0.3647059 0.3960784 0.3803922
##  [841] 0.3686275 0.3921569 0.3960784 0.3568627 0.3803922 0.3254902
##  [847] 0.3843137 0.3921569 0.3960784 0.3803922 0.3882353 0.3725490
##  [853] 0.3921569 0.3647059 0.3882353 0.3960784 0.3960784 0.3921569
##  [859] 0.3294118 0.3686275 0.3764706 0.3372549 0.3882353 0.3921569
##  [865] 0.3960784 0.3882353 0.3843137 0.3803922 0.3921569 0.3843137
##  [871] 0.3843137 0.3921569 0.3764706 0.3686275 0.3921569 0.3803922
##  [877] 0.3803922 0.3725490 0.3843137 0.3960784 0.3764706 0.3921569
##  [883] 0.3803922 0.3960784 0.3568627 0.3686275 0.3686275 0.3254902
##  [889] 0.3568627 0.3843137 0.3882353 0.3882353 0.3843137 0.3725490
##  [895] 0.3803922 0.3764706 0.3764706 0.3843137 0.3843137 0.3411765
##  [901] 0.3803922 0.3921569 0.3764706 0.3764706 0.3607843 0.3333333
##  [907] 0.3411765 0.3843137 0.3843137 0.3568627 0.3843137 0.3372549
##  [913] 0.3058824 0.3686275 0.3411765 0.3411765 0.3137255 0.3568627
##  [919] 0.3960784 0.3960784 0.3607843 0.3725490 0.3647059 0.3529412
##  [925] 0.3647059 0.3843137 0.3921569 0.3843137 0.3529412 0.3803922
##  [931] 0.3960784 0.3843137 0.3921569 0.3686275 0.3647059 0.3607843
##  [937] 0.3647059 0.3490196 0.3529412 0.3215686 0.2784314 0.3058824
##  [943] 0.3254902 0.3215686 0.3058824 0.3058824 0.3294118 0.3372549
##  [949] 0.3215686 0.3176471 0.3372549 0.3411765 0.3490196 0.3686275
##  [955] 0.3450980 0.3568627 0.3843137 0.3607843 0.3647059 0.3882353
##  [961] 0.3921569 0.3568627 0.3764706 0.3921569 0.3921569 0.3725490
##  [967] 0.3960784 0.3960784 0.3764706 0.3607843 0.3764706 0.3725490
##  [973] 0.3725490 0.3960784 0.3921569 0.3921569 0.3921569 0.3411765
##  [979] 0.3254902 0.3372549 0.3647059 0.3960784 0.3960784 0.3921569
##  [985] 0.3725490 0.3921569 0.3960784 0.3960784 0.3843137 0.3960784
##  [991] 0.3568627 0.3882353 0.3921569 0.3843137 0.3960784 0.3686275
##  [997] 0.3960784 0.3921569 0.3725490 0.3921569 0.3882353 0.3803922
## [1003] 0.3843137 0.3882353 0.3764706 0.3921569 0.3921569 0.3686275
## [1009] 0.3764706 0.3568627 0.3686275 0.3960784 0.3843137 0.3647059
## [1015] 0.3882353 0.3568627 0.3411765 0.3960784 0.3843137 0.3607843
## [1021] 0.3803922 0.3882353 0.3921569 0.3764706 0.3490196 0.3568627
## [1027] 0.3764706 0.3607843 0.3843137 0.3647059 0.3372549 0.3450980
## [1033] 0.3764706 0.3921569 0.3921569 0.3647059 0.3490196 0.3647059
## [1039] 0.3568627 0.3529412 0.3764706 0.3960784 0.3803922 0.3921569
## [1045] 0.3568627 0.3529412 0.3764706 0.3960784 0.3960784 0.3882353
## [1051] 0.3882353 0.3803922 0.3725490 0.3843137 0.3803922 0.3960784
## [1057] 0.3725490 0.3803922 0.3764706 0.3725490 0.3960784 0.3764706
## [1063] 0.3960784 0.3803922 0.3411765 0.3333333 0.3568627 0.3960784
## [1069] 0.3803922 0.3960784 0.3803922 0.3686275 0.3921569 0.3882353
## [1075] 0.3960784 0.3725490 0.3843137 0.3803922 0.3568627 0.3647059
## [1081] 0.3490196 0.3254902 0.3098039 0.3333333 0.3529412 0.3607843
## [1087] 0.3764706 0.3960784 0.3686275 0.3529412 0.3882353 0.3882353
## [1093] 0.3568627 0.3294118 0.3098039 0.3294118 0.3568627 0.3450980
## [1099] 0.3215686 0.2980392 0.2980392 0.3294118 0.3764706 0.3843137
## [1105] 0.3921569 0.3882353 0.3921569 0.3686275 0.3529412 0.3490196
## [1111] 0.3607843 0.3960784 0.3686275 0.3529412 0.3882353 0.3960784
## [1117] 0.3921569 0.3921569 0.3686275 0.3725490 0.3921569 0.3725490
## [1123] 0.3843137 0.3803922 0.3882353 0.3803922 0.3960784 0.3921569
## [1129] 0.3921569 0.3960784 0.3764706 0.3294118 0.3098039 0.3686275
## [1135] 0.3843137 0.3764706 0.3882353 0.3960784 0.3960784 0.3843137
## [1141] 0.3725490 0.3882353 0.3882353 0.3725490 0.3568627 0.3529412
## [1147] 0.3647059 0.3725490 0.3882353 0.3607843 0.3450980 0.3607843
## [1153] 0.3921569 0.3803922 0.3725490 0.3960784 0.3725490 0.3607843
## [1159] 0.3725490 0.3960784 0.3960784 0.3882353 0.3921569 0.3960784
## [1165] 0.3921569 0.3607843 0.3294118 0.3490196 0.3686275 0.3686275
## [1171] 0.3490196 0.3490196 0.3686275 0.3803922 0.3843137 0.3960784
## [1177] 0.3921569 0.3960784 0.3921569 0.3921569 0.3921569 0.3725490
## [1183] 0.3882353 0.3607843 0.3529412 0.3843137 0.3803922 0.3882353
## [1189] 0.3960784 0.3960784 0.3960784 0.3725490 0.3686275 0.3843137
## [1195] 0.3882353 0.3960784 0.3921569 0.3803922 0.3921569 0.3725490
## [1201] 0.3764706 0.3607843 0.3921569 0.3921569 0.3921569 0.3843137
## [1207] 0.3686275 0.3764706 0.3843137 0.3921569 0.3960784 0.3960784
## [1213] 0.3882353 0.3921569 0.3921569 0.3921569 0.3803922 0.3882353
## [1219] 0.3882353 0.3803922 0.3803922 0.3882353 0.3960784 0.3529412
## [1225] 0.3607843 0.3843137 0.3764706 0.3725490 0.3450980 0.3764706
## [1231] 0.3882353 0.3960784 0.3803922 0.3607843 0.3764706 0.3960784
## [1237] 0.3843137 0.3921569 0.3803922 0.3686275 0.3843137 0.3960784
## [1243] 0.3921569 0.3882353 0.3882353 0.3960784 0.3843137 0.3686275
## [1249] 0.3803922 0.3960784 0.3921569 0.3764706 0.3686275 0.3921569
## [1255] 0.3568627 0.3490196 0.3921569 0.3960784 0.3764706 0.3490196
## [1261] 0.3725490 0.3960784 0.3764706 0.3607843 0.3568627 0.3607843
## [1267] 0.3960784 0.3843137 0.3921569 0.3921569 0.3725490 0.3882353
## [1273] 0.3960784 0.3764706 0.3725490 0.3764706 0.3960784 0.3960784
## [1279] 0.3882353 0.3843137 0.3843137 0.3921569 0.3960784 0.3725490
## [1285] 0.3647059 0.3647059 0.3960784 0.3882353 0.3843137 0.3882353
## [1291] 0.3921569 0.3960784 0.3921569 0.3647059 0.3764706 0.3803922
## [1297] 0.3960784 0.3960784 0.3843137 0.3803922 0.3921569 0.3764706
## [1303] 0.3803922 0.3921569 0.3843137 0.3843137 0.3882353 0.3764706
## [1309] 0.3647059 0.3882353 0.3921569 0.3960784 0.3960784 0.3960784
## [1315] 0.3686275 0.3882353 0.3803922 0.3725490 0.3882353 0.3960784
## [1321] 0.3882353 0.3921569 0.3960784 0.3921569 0.3843137 0.3686275
## [1327] 0.3725490 0.3490196 0.3647059 0.3921569 0.3921569 0.3882353
## [1333] 0.3960784 0.3568627 0.3803922 0.3960784 0.3725490 0.3725490
## [1339] 0.3333333 0.3882353 0.3921569 0.3764706 0.3725490 0.3882353
## [1345] 0.3686275 0.3882353 0.3960784 0.3960784 0.3803922 0.3843137
## [1351] 0.3921569 0.3568627 0.3764706 0.3490196 0.3803922 0.3960784
## [1357] 0.3960784 0.3921569 0.3882353 0.3960784 0.3803922 0.3686275
## [1363] 0.3960784 0.3882353 0.3960784 0.3686275 0.3450980 0.3568627
## [1369] 0.3960784 0.3686275 0.3725490 0.3843137 0.3725490 0.3529412
## [1375] 0.3333333 0.3411765 0.3843137 0.3921569 0.3921569 0.3882353
## [1381] 0.3803922 0.3803922 0.3764706 0.3764706 0.3882353 0.3960784
## [1387] 0.3843137 0.3843137 0.3960784 0.3960784 0.3921569 0.3647059
## [1393] 0.3686275 0.3803922 0.3843137 0.3960784 0.3764706 0.3647059
## [1399] 0.3843137 0.3607843 0.3294118 0.3450980 0.3647059 0.3843137
## [1405] 0.3490196 0.3058824 0.3215686 0.3294118 0.3882353 0.3921569
## [1411] 0.3921569 0.3960784 0.3921569 0.3372549 0.3725490 0.3843137
## [1417] 0.3803922 0.3843137 0.3921569 0.3921569 0.3529412 0.3686275
## [1423] 0.3843137 0.3921569 0.3960784 0.3764706 0.3490196 0.3529412
## [1429] 0.3921569 0.3725490 0.3372549 0.3058824 0.3176471 0.3568627
## [1435] 0.3843137 0.3725490 0.3764706 0.3921569 0.3960784 0.3843137
## [1441] 0.3254902 0.3254902 0.3411765 0.3725490 0.3921569 0.3843137
## [1447] 0.3960784 0.3686275 0.3647059 0.3803922 0.3803922 0.3843137
## [1453] 0.3843137 0.3843137 0.3529412 0.3333333 0.3568627 0.3960784
## [1459] 0.3803922 0.3921569 0.3882353 0.3764706 0.3764706 0.3921569
## [1465] 0.3960784 0.3686275 0.3411765 0.3725490 0.3843137 0.3803922
## [1471] 0.3843137 0.3921569 0.3803922 0.3882353 0.3882353 0.3882353
## [1477] 0.3764706 0.3960784 0.3882353 0.3960784 0.3882353 0.3529412
## [1483] 0.3921569 0.3921569 0.3960784 0.3647059 0.3882353 0.3960784
## [1489] 0.3607843 0.3764706 0.3686275 0.3803922 0.3803922 0.3921569
## [1495] 0.3568627 0.3490196 0.3843137 0.3686275 0.3490196 0.3764706
## [1501] 0.3843137 0.3529412 0.3803922 0.3882353 0.3764706 0.3921569
## [1507] 0.3529412 0.3921569 0.3803922 0.3960784 0.3882353 0.3921569
## [1513] 0.3960784 0.3921569 0.3882353 0.3960784 0.3803922 0.3843137
## [1519] 0.3960784 0.3882353 0.3686275 0.3568627 0.3803922 0.3921569
## [1525] 0.3960784 0.3725490 0.3568627 0.3882353 0.3843137 0.3882353
## [1531] 0.3960784 0.3960784 0.3921569 0.3921569 0.3960784 0.3960784
## [1537] 0.3882353 0.3490196 0.3921569 0.3803922 0.3921569 0.3843137
## [1543] 0.3843137 0.3529412 0.3647059 0.3607843 0.3921569 0.3843137
## [1549] 0.3882353 0.3960784 0.3960784 0.3725490 0.3647059 0.3960784
## [1555] 0.3960784 0.3764706 0.3764706 0.3882353 0.3882353 0.3803922
## [1561] 0.3725490 0.3764706 0.3882353 0.3960784 0.3960784 0.3764706
## [1567] 0.3764706 0.3803922 0.3921569 0.3960784 0.3960784 0.3921569
## [1573] 0.3882353 0.3843137 0.3921569 0.3921569 0.3647059 0.3647059
## [1579] 0.3686275 0.3843137 0.3686275 0.3725490 0.3921569 0.3960784
## [1585] 0.3803922 0.3725490 0.3647059 0.3764706 0.3921569 0.3764706
## [1591] 0.3882353 0.3882353 0.3882353 0.3921569 0.3725490 0.3294118
## [1597] 0.3254902 0.3294118 0.3450980 0.3843137 0.3725490 0.3725490
## [1603] 0.3450980 0.3490196 0.3843137 0.3960784 0.3725490 0.3960784
## [1609] 0.3843137 0.3921569 0.3607843 0.3450980 0.3803922 0.3882353
## [1615] 0.3725490 0.3647059 0.3568627 0.3764706 0.3882353 0.3843137
## [1621] 0.3960784 0.3960784 0.3803922 0.3960784 0.3882353 0.3607843
## [1627] 0.3764706 0.3960784 0.3921569 0.3960784 0.3843137 0.3882353
## [1633] 0.3843137 0.3960784 0.3921569 0.3607843 0.3568627 0.3686275
## [1639] 0.3686275 0.3882353 0.3725490 0.3647059 0.3843137 0.3490196
## [1645] 0.3882353 0.3882353 0.3647059 0.3647059 0.3882353 0.3960784
## [1651] 0.3725490 0.3843137 0.3960784 0.3921569 0.3725490 0.3686275
## [1657] 0.3960784 0.3843137 0.3764706 0.3529412 0.3490196 0.3254902
## [1663] 0.3450980 0.3882353 0.3960784 0.3764706 0.3725490 0.3686275
## [1669] 0.3725490 0.3843137 0.3725490 0.3294118 0.3411765 0.3529412
## [1675] 0.3411765 0.3490196 0.3568627 0.3372549 0.3254902 0.3647059
## [1681] 0.3921569 0.3843137 0.3882353 0.3882353 0.3764706 0.3686275
## [1687] 0.3647059 0.3333333 0.3098039 0.3450980 0.3803922 0.3882353
## [1693] 0.3803922 0.3803922 0.3882353 0.3803922 0.3725490 0.3607843
## [1699] 0.3529412 0.3725490 0.3607843 0.3529412 0.3921569 0.3921569
## [1705] 0.3960784 0.3568627 0.3686275 0.3764706 0.3921569 0.3647059
## [1711] 0.3647059 0.3764706 0.3411765 0.3647059 0.3568627 0.3215686
## [1717] 0.3607843 0.3607843 0.3960784 0.3921569 0.3411765 0.3294118
## [1723] 0.3333333 0.3372549 0.3254902 0.3254902 0.3764706 0.3607843
## [1729] 0.3764706 0.3764706 0.3921569 0.3882353 0.3764706 0.3254902
## [1735] 0.3450980 0.3921569 0.3960784 0.3843137 0.3960784 0.3960784
## [1741] 0.3882353 0.3921569 0.3882353 0.3882353 0.3843137 0.3803922
## [1747] 0.3764706 0.3450980 0.3647059 0.3960784 0.3764706 0.3843137
## [1753] 0.3764706 0.3607843 0.3607843 0.3725490 0.3921569 0.3411765
## [1759] 0.3215686 0.3607843 0.3843137 0.3568627 0.3450980 0.3372549
## [1765] 0.3411765 0.3686275 0.3764706 0.3843137 0.3843137 0.3882353
## [1771] 0.3882353 0.3647059 0.3725490 0.3647059 0.3215686 0.3176471
## [1777] 0.3529412 0.3803922 0.3921569 0.3960784 0.3843137 0.3803922
## [1783] 0.3803922 0.3882353 0.3725490 0.3882353 0.3921569 0.3960784
## [1789] 0.3803922 0.3921569 0.3882353 0.3843137 0.3960784 0.3882353
## [1795] 0.3882353 0.3960784 0.3882353 0.3686275 0.3921569 0.3450980
## [1801] 0.3647059 0.3882353 0.3843137 0.3921569 0.3960784 0.3882353
## [1807] 0.3803922 0.3882353 0.3803922 0.3921569 0.3921569 0.3843137
## [1813] 0.3921569 0.3882353 0.3686275 0.3960784 0.3686275 0.3607843
## [1819] 0.3725490 0.3882353 0.3803922 0.3803922 0.3960784 0.3882353
## [1825] 0.3921569 0.3607843 0.3529412 0.3450980 0.3686275 0.3843137
## [1831] 0.3803922 0.3960784 0.3921569 0.3960784 0.3764706 0.3568627
## [1837] 0.3529412 0.3921569 0.3882353 0.3882353 0.3882353 0.3529412
## [1843] 0.3647059 0.3843137 0.3882353 0.3686275 0.3921569 0.3921569
## [1849] 0.3882353 0.3647059 0.3647059 0.3921569 0.3803922 0.3686275
## [1855] 0.3725490 0.3686275 0.3843137 0.3960784

Changing the out-of-limit pixel values to 0 (black).

part2img[part2img > upperlimit] <- 0
part2img[part2img < lowerlimit] <- 0

Display of the new image and the original image in a plot.

part2img2 <- readJPEG("C:/Users/ilker zeybek/Desktop/423part1grayscale.jpg", native = FALSE)
plot(NA,xlim=c(0,nrow(part2img)),ylim=c(0,ncol(part2img)))
rasterImage(part2img,0,0,nrow(part2img),ncol(part2img))

plot(NA,xlim=c(0,nrow(part2img2)),ylim=c(0,ncol(part2img2)))
rasterImage(part2img2,0,0,nrow(part2img2),ncol(part2img2))

Since our image is grayscale, we have expected that our image has more pixel values below the lower limit than pixel values over the upper limit. It is clearly shown that it has significantly more pixel values below lower limit since there is not any pixel value over the upper limit. Pixel values lower than the lower limit is on the darker colored part of the original grayscale image of table because their pixel values are closer to 0 than brighter colored parts of the table. Only explanation that we do not have any pixel values over the upper limit is image of the table is grayscale that doesn’t has any colored parts like in the original image used in part 1.

4. We have created loop for detecting 51x51 parts of the grayscale image and found mean and standard deviation of each 51x51 window. Calculated upper and lower limits for changing pixel values for each window. Used ifelse() function for detecting out-of-bound pixels and changed them with 0 (black) pixel value.

fracturepic <- part2img2
for (i in 1:10){
  for(j in 1:10){
    fiftyonepatch <- fracturepic[(51*(i-1)+1):(51*i),(51*(j-1)+1):(51*j)]
    meanpatch <- mean(fiftyonepatch)
    stdpatch <- sd(fiftyonepatch)
    patchupperlimit <- qnorm(0.999,meanpatch,stdpatch)
    patchlowerlimit <- qnorm(0.001,meanpatch,stdpatch)
    fracturepic[(51*(i-1)+1):(51*i),(51*(j-1)+1):(51*j)] <- ifelse(fracturepic[(51*(i-1)+1):(51*i),(51*(j-1)+1):(51*j)] < patchlowerlimit,0,fracturepic[(51*(i-1)+1):(51*i),(51*(j-1)+1):(51*j)])
    fracturepic[(51*(i-1)+1):(51*i),(51*(j-1)+1):(51*j)] <- ifelse(fracturepic[(51*(i-1)+1):(51*i),(51*(j-1)+1):(51*j)] > patchupperlimit,0,fracturepic[(51*(i-1)+1):(51*i),(51*(j-1)+1):(51*j)])}}

Display of 51x51 patched and Original Image in a single plot

par(mfrow=c(1,2))
plot(NA,xlim=c(0,nrow(fracturepic)),ylim=c(0,ncol(fracturepic)),xlab="Horizontal",ylab="Vertical")
rasterImage(fracturepic,0,0,nrow(fracturepic),ncol(fracturepic))
plot(NA,xlim=c(0,nrow(part2img)),ylim=c(0,ncol(part2img)),xlab="Horizontal",ylab="Vertical")
rasterImage(part2img2,0,0,nrow(part2img2),ncol(part2img2))

We expected more black pixels since we have treated our original grayscale image like 100 different 51x51 sized pictures. Each window has less population size and its own boundary limits. This caused more black pixels in the patched image.